Stable Kalman Filter estimator with given covariance matrices

I asked this question a while back. Essentially considering the follow basic Kalman Filter, following the Wikipedia convention.

$$$$\begin{split} x_k &= F_kx_{k-1} + B_k u_k +w_k\\ y_k &= H_kx_k + v_k \end{split}$$$$

Where $$w_k \sim \mathcal{N}(0, Q)$$, $$v_k \sim \mathcal{N}(0, R)$$. I wonder how critical the estimation of $$R$$ and $$Q$$ are.

Assuming that your model of the dynamics is correct (i.e. $$F_k$$, $$B_k$$ and $$H_k$$) then the good news is that any (positive semi-definite) choice for $$Q$$ and $$R$$ will result in a state estimator that is stable, so the estimated state converges to the true state in the absence of disturbances $$w_k$$ and $$v_k$$ and converge to some bound around the true state when there are disturbances.

Can I understand why any positive definite (to avoid edge cases, let's assume positive definite rather than positive semidefinite) $$R$$ and $$Q$$ will result in a stable estimator.

Looking at the asymptotic form,

$$\mathbf{P}_\infty = \mathbf{F}\left(\mathbf{P}_\infty - \mathbf{P}_\infty \mathbf{H}^\textsf{T} \left(\mathbf{H}\mathbf{P}_\infty\mathbf{H}^\textsf{T} + \mathbf{R}\right) ^{-1} \mathbf{H}\mathbf{P}_\infty\right) \mathbf{F}^\textsf{T} + \mathbf{Q}$$

$$\mathbf{K}_\infty = \mathbf{P}_\infty \mathbf{H}^\textsf{T} \left( \mathbf{R} + \mathbf{H} \mathbf{P}_\infty \mathbf{H}^\textsf{T} \right) ^{-1}.$$

and finally

$$\hat{x}_{k+1}=(F-\overline{K} _{\infty}H){\hat {x} }_{k} + B u_k +\overline{K} _{\infty} {z} _{k}$$

$$\hat{x}$$ still depends on $$R$$ and $$Q$$. Why are we saying it will converge to true state, as long as $$R$$ and $$Q$$ are positive definite?

• I don't know the answer off the top of my head, but Dan Simon's book "Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches" has a good discussion about Kalman Filter stability in Chapter 7. Commented May 27 at 17:43

A short stability proof of the Kalman filter is given in section 2.4.1 of:

Rhudy, M. B. (2013). Sensitivity and stability analysis of nonlinear Kalman filters with application to aircraft attitude estimation. West Virginia University.

I will give a short summary of what is stated in this dissertation. First of all it is assumed that $$Q$$ and $$R$$ (these two matrices can also be time varying) are such that $$(F_k,Q)$$ is uniformly controllable and $$(F_k,H_k^\top R^{-1} H_k)$$ is uniformly observable, then the error covariance matrix $$P_k$$ has a finite positive definite upper and lower bound.

The a priori estimation error $$\tilde{x}_k = \hat{x}_k - x_k$$ can shown to be stable using the following Lyapunov function

$$V(\tilde{x}_k) = \tilde{x}_k^\top P_k^{-1} \tilde{x}_k,$$

which is guaranteed to be a positive definite function when the controllability and observability conditions are satisfied. The estimation error dynamics is given by

$$\tilde{x}_{k+1} = F_k\,(I - K_k\,H_k)\,\tilde{x}_{k} + F_k\,K_k\,v_k - w_k.$$

For the bounded input bounded output (BIBO) stability analysis it is only required to consider the homogenous part of the error dynamics. Doing so yields to following increment of the proposed Lyapunov function

\begin{align} \Delta V(\tilde{x}_k) &= \tilde{x}_{k+1}^\top P_{k+1}^{-1} \tilde{x}_{k+1} - \tilde{x}_k^\top P_k^{-1} \tilde{x}_k, \\ &= \tilde{x}_k^\top \left[(I - K_k\,H_k)^\top F_k^\top P_{k+1}^{-1} F_k (I - K_k\,H_k) - P_k^{-1}\right] \tilde{x}_k, \end{align}

which after some more manipulation can shown to be negative definite. Previously it was shown that $$P_k$$ is upper and lower bounded and therefore $$P_k^{-1}$$ should be as well, which together with $$\Delta V(\tilde{x}_k)\prec 0$$ enables one to show that the Kalman filter is BIBO stable.

For showing that the Kalman filter in stable one only needs that $$w_k$$ and $$v_k$$ are bounded, so nowhere is it required that $$Q$$ and and $$R$$ are such that $$w_k \sim \mathcal{N}(0, Q)$$ and $$v_k \sim \mathcal{N}(0, R)$$. So the stochastic terms can have very different covariance matrices associated with them. Since the Kalman filter is dealing with a stochastic system an more in-depth analysis is needed to show that covariance of the error remains bounded when there is a mismatch between noise covariance matrices the actual noise compared to the ones used in the Kalman filter. Further resources regarding this are mentioned earlier mentioned dissertation.

However, having a mismatch in the noise covariance matrices can result in a far from optimal noise suppression, as is discussed in:

Ge, Q., Shao, T., Duan, Z., & Wen, C. (2016). Performance analysis of the Kalman filter with mismatched noise covariances. IEEE Transactions on Automatic Control, 61(12), 4014-4019.

• Thanks. What's incorrect about my understanding though. It seems like in the asymptotic form, we still have $R$ and $Q$ in $\hat{x}_{k+1}$. I am not sure why that suggests we convert to true state (which I think it means $|\hat{x}_{k} - x_k| \rightarrow 0$? Commented May 28 at 15:20
• @MattFrank the Kalman filter is stable for any choice of $Q$ and $R$, as long as the controllability and observability conditions are met. Commented May 28 at 22:58
• @KwinvanderVeen showing $\Delta V(\tilde{x}_k)\prec 0$ is not enough for stability in this form. You need to show $\Delta V(\tilde{x}_k)\prec -\eta I$ for some constant $\eta > 0$. Commented Jun 4 at 12:16